It has recently become popular to render various kinds of three-dimensional objects on a two-dimensional display screen with a computer graphics system in research or engineering applications. One of these applications is molecular modeling or molecular simulation which is used to research characteristics of substances by displaying a three-dimensional molecular model on a display such as a CRT, a printer, or a plotter. In molecular modeling, spherical and curved surfaces are three-dimensionally depicted by using a plurality of dots on a two-dimensional display screen.
The method of rendering a three-dimensional object by using a plurality of dots is simple compared with the so-called solid model method or space filling representation in which spherical and curved surfaces are completely smeared or painted while maintaining the three-dimensional feeling. Therefore, an advantage exists in that rendering can be executed at a high speed without requiring a great deal of computer throughput. The present invention relates to a method of and an apparatus for rendering three-dimensional objects, including a body of revolution, such as a sphere, by using a plurality of dots.
Throughout this specification, a dot may also be referred to as a point, a marker or any other shapes. They have a limited size and are adequately small when compared with the entire size of a rendered three-dimensional object. Shape of a dot, marker, or point may include lines, circles, crosses, or any other shape. They may be displayed as luminescent points, black points, or in any other color on the display screen.
The importance of depicting the spherical surfaces of a molecular model by using a plurality of dots is described by Bash, P. A. et al. in "Van der Waals Surfaces in Molecular Modeling, Implementation with Real Time Computer Graphics," Science, Vol. 222, pp. 1325-1327, 1983.
The conventional method of three-dimensionally depicting spherical surfaces by using a plurality of dots includes the flat-to-spherical-surface equal-area mapping method (for details, see "Guide to Molecular Simulation" written by A. Koide, edited by Okada and Osawa, and published by Kaibundo, pp. 209-210, 1989). This prior art method is shown in FIG. 10. Lattice 100 of equilateral triangles on a horizontal plane is projected in parallel with a projection direction A onto a sphere 101. In this method, the sphere 101 is represented by dots placed on the surface of the sphere 101 corresponding to the lattice points 100 or the apexes of equilateral triangles on the plane. The drawback of this prior art method is that, although the view of the sphere 101 from the projection direction A is good, the view of the sphere 101 from the direction B which is perpendicular to the projection direction A is not good because dots are concentrated on a circumference area 103 of the sphere 101 in view from the projection direction A. And when the sphere 101 is rotated, the circumference area 103 seems to be a ring or a stripe on the sphere 101 because dots are more concentrated on the circumference area 103 than any other surface areas in the sphere 101.
Another prior art method of rendering a sphere by using a plurality dots is the latitude-longitude intersection method used in "Molecular Design Support System," program Nos. 5788- JKF/JKG/JKH/JKJ/JKL/JKN, which is developed by IBM-Japan and which is one of the graphics and program products of IBM-Japan. This prior art method is shown in FIG. 11. In this prior art method, a sphere 111 is rendered by using a plurality of dots which are placed at intersections 112 between latitudes 113 and longitudes 114 on the surface of the sphere 111. The drawback of this prior art method is that dots are concentrated on polar areas 115, 116 of the sphere 111. On the other hand, the dots are scarce at the equatorial area of the sphere 111.
The other method of rendering a sphere by using a plurality dots is called the point symmetry group method. This method uses symmetry technique as shown in FIG. 12. If one dot 121 at Cartesian coordinates (x,y,z) on a sphere 120 is determined, 47 other dots on the sphere 120 are determined at the following Cartesian coordinates: ##EQU1##
FIG. 13 shows the dot representation of a sphere by using this prior art method. This method is used in ANCHOR, a three dimensional molecular design support package, of Fujitsu, in which a dot model for a molecular is displayed by a plurality of dots by using this prior art method. The drawback of this prior art method is that there is no pole or orientation of the sphere as shown in FIG. 13 rendered by using this method. Therefore, it is difficult to tell the orientation of a sphere depicted by using this prior art method after rotation or revolution of the sphere.
In the prior art equal-area mapping method as shown in FIG. 10, a view of a three-dimensional model from an angle different from the projection direction while continuously, that is, in real time, rotating the model is not preferable. This is because the region of outer periphery of the spherical surface in view from the projection direction where dots are concentrated appears as a stripe or a ring similar to the equator. In the prior art latitude-longitude intersection method as shown in FIG. 11, it is occasionally difficult to observe the model because dots tend to concentrate at the north and south polar regions and disperse around the equatorial region. In the prior art point symmetry group method as shown in FIG. 13, more dots are generally required for rendering and more computer throughput is required for the calculation of coordinate conversion of dots to rotate the molecular model. In addition, the dots after revolution or rotation of the sphere are unnaturally arranged. Thus, it is difficult to tell the orientation of a sphere after rotation or revolution of the sphere rendered by using this prior art method.